standard 12 basic polygon definitions ... - whs...

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Basic Polygon Definitions

Interior Angle Sum Theorem

Exterior Angle Sum Theorem

PROBLEM 1a PROBLEM 1b





Standard 12

END SHOWPRESENTATION CREATED BY SIMON PEREZ. All rights reserved

http://www.mrperezonlinemathtutor.com/

2

Standard 12:

Students find and use measures of sides, interior and exterior

angles of triangles and polygons to classify figures and solve

problems.

Estándar 12:

Los estudiantes encuentran y usan medidas de los lados,

ángulos interiores y exteriores de triángulos y polígonos para

clasificar figuras y resolver problemas.

PRESENTATION CREATED BY SIMON PEREZ. All rights reserved


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These are examples of POLYGONS:

These are NOT POLYGONS:

What is the difference? Click to find out…

Standard 12



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Definition of a POLYGON:

A polygon is closed figure formed by a finite number of coplanar segments such

that

1.- the sides that have a common endpoint are noncollinear, and

2.- each side intersects exactly two other sides, but only at their endpoints.

Standard 12

Do you know what is a Convex Polygon? Click to find out…



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CONVEX POLYGON:

None of the lines containing a side of

the polygon contains a point inside

the polygon.

NON-CONVEX POLYGON OR

CONCAVE POLYGON:

One or more lines containing a

side on the polygon, contain

points inside the polygon.

Standard 12

How do we call a polygon with 10 sides? Click to find out…



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Number of sides Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

N ngon

Standard 12

What is a regular polygon? Click to find out…



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Definition of Regular Polygon:

A regular polygon is a convex polygon with all sides congruent and

all angles congruent.

Standard 12



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A B

CD

What is the sum of the interior angles in the quadrilateral below?

We draw diagonal AC

then we have ABC ADCand

1 2

3numbering all angles in ABC45

6

ADCnumbering all angles in

then

m 1 m 2 m 3+ + = 180°

m 4 m 5 m 6+ + = 180°

Because the sum of the interior

angles of a triangle is 180°

adding both equations

m 1 m 2 m 3+ + + 180°m 4 m 5 m 6+ + = 180°+

Rearrenging the terms

m 1 m 6 m 2+ + + 180°m 3 m 4 m 5+ + = 180°+

m A m B m C m D+ + + = 2(180°) or 360°

Could this be done for a Pentagon, an hexagon, etc.? Click to find out…

Standard 12



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What would be the sum of the interior angles for all these polygons?

2(180°) or 360° 3(180°) or 540° 4(180°) or 720° 5(180°) or 900° 6(180°) or 1080°

Can you see a pattern? How this relates to the number of sides?

4 sides 5 sides 6 sides 7 sides8 sides

4-2 5-2 6-2 7-2 8-2

(number of sides-2)180°=Sum of interior angles in the polygon

Standard 12

How can we define this? Click to find out…



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Interior angle sum theorem

If a convex polygon has n sides and S is the sum of the measures of the

interior angles, then:

S=180°(n-2)

Standard 12



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Find the sum of the measures of the interior angles of a convex octagon.

We know that a Octagon is a polygon with 8 sides, so n=8

andS=180°(n-2)

Where S= Sum of measures of interior angles

So:S=180°(8-2)

S=180°(6)

S=1080°

Standard 12



12

Find the sum of the measures of the interior angles of a convex decagon.

We know that a Decagon is a polygon with 10 sides, so n=10

andS=180°(n-2)

Where S= Sum of measures of interior angles

So:S=180°(10-2)

S=180°(8)

S=1440°

Standard 12



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Find the measure of each angle:

110°(8X+6)°

(6X-4)°

(10x+8)°(8X)°

(2X)°

We know from the Interior Angle Sum Theorem that:

S=180°(n-2)

So: 110° + (2X)° + (8X)° + (10X+8)° + (6X-4)° + (8X+6)°= 180°(6-2)

There are 6 sides so: n=6

2X + 8X + 10X + 6X + 8X + 110 +8 -4 +6 = 180(4)

34X + 120 = 720

-120 -120

34X = 60034 34

X 17.6

Then:

8X+6 = 8( )+617.6

146.8°

6X-4= 6( )-417.6

10X+8= 10( )+817.6

8X= 8( )17.6

2X= 2( )17.6

35.2°

35.2°

140.8° 184°

101.6°

146.8°

S =110° + (2X)° + (8X)° + (10X+8)° + (6X-4)° + (8X+6)°

and

Checking: 146.8° + 101.6°+184°+140.8°+35.2°+110° 720°

Standard 12

140.8°

101.6°

184°



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Find the measure of each angle:

100°(7X-9)°

(10X+3)°

(9X-6)°(10X)°

(7X)°

We know from the Interior Angle Sum Theorem that:

S=180°(n-2)

So: 100° + (7X)° + (10X)° + (9X-6)° + (10X+3)° + (7X-9) = 180°(6-2)

There are 6 sides so: n=6

7X + 10X + 9X + 10X + 7X + 100 – 6 +3 -9 = 180(4)

43X + 88 = 720

-88 -88

43X = 63243 43

X 14.7

Then:

7X-9 = 7( )-914.7

93.9°

10X+3= 10( )+3 14.7

9X-6= 9( )-614.7

126.3°

10X= 10( )14.7

7X= 7( )14.7

102.9°

102.9°

147° 126.3°

150°

93.9°

S =100° + (7X)° + (10X)° + (9X-6)° + (10X+3)° + (7X-9)°

and

Checking: 93.9° + 150°+126.3°+147°+102.9°+100° 720°

Standard 12

147°

150°



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These are all the EXTERIOR ANGLES for the

polygon at the right.

m 1 m 2+ = 180°

Each exterior angle is supplementry to its interior angle.

1

1

2

Angles and form a linear

pair2

Standard 12

What is the sum of the exterior angles in a polygon if we know the sum of the

interior angles? Click to find out



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For each vertex we have a linear pair, so:

measure of EXTERIOR ANGLE + measure of INTERIOR ANGLE =180°

substracting measure of INTERIOR ANGLE from both sides:

measure of EXTERIOR ANGLE=180°-measure of INTERIOR ANGLE

Multiplying both sides for n=number of vertices:

n(measure of EXTERIOR ANGLE)=n(180°-measure of INTERIOR ANGLE)

n(measure of EXTERIOR ANGLE)=n180°-n(measure of INTERIOR ANGLE)

n180° -EXTERIOR ANGLE SUM = INTERIOR ANGLE SUM

=EXTERIOR ANGLE SUM n180° - 180°(n-2)

EXTERIOR ANGLE SUM = n180°-n180°+360°

EXTERIOR ANGLE SUM= 360°

Standard 12

Click for a formal definition…

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Exterior Angle Sum Theorem:

If a polygon is convex, then the sum of the measures of the exterior angles, one at

each vertex, is 360°.

Standard 12



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An exterior angle of a regular polygon is 45°. Find the number of sides.

From the Exterior Angle Sum Theorem, we know that the sum of the exterior angles

is 360°, so:

360°

45°=8 So we have 8 vertices, and 8 Sides.

1

8

2

4 3

6

5

7

Standard 12



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An exterior angle of a regular polygon is 72°. Find the number of sides.

From the Exterior Angle Sum Theorem, we know that the sum of the exterior angles

is 360°, so:

360°

72°=5 So we have 5 vertices, and 5 Sides.

1

2

4

3

5

Standard 12



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Find the number of sides in a regular polygon if the measure of each interior angle

is 165°.

We know that in a vertex in the polygon, the interior angle forms a linear pair with

the exterior angle. So:




=180°-165°

=15°

And from Exterior Angle Sum Theorem, the sum of all exterior angles is 360°, so:

360°

15°=24 There are 24 vertices and thus 24 sides.

Standard 12



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Find the number of sides in a regular polygon if the measure of each interior angle

is 156°.

We know that in a vertex in the polygon, the interior angle forms a linear pair with

the exterior angle. So:




=180°-156°

=24°

And from Exterior Angle Sum Theorem, the sum of all exterior angles is 360°, so:

360°

24°=15 There are 15 vertices and thus 15 sides.

Standard 12



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Find the measures of each interior and each exterior angle of a regular 60-gon?

First we know that in a regular n-gon all interior and exterior angles are congruent

and the sum of exterior angles is 360°, so:

360°

60 vertices= measure of EXTERIOR ANGLE

360°

60=6°

And the interior and exterior angles form a linear pair, so:


substracting measure of EXTERIOR ANGLE from both sides:

measure of INTERIOR ANGLE=180°-measure of EXTERIOR ANGLE

=180°-6°

So: measure of EXTERIOR ANGLE = 6°

measure of INTERIOR ANGLE=174° Standard 12



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Find the measures of each interior and each exterior angle of a regular 24-gon?

First we know that in a regular n-gon all interior and exterior angles are congruent

and the sum of exterior angles is 360°, so:

360°

24 vertices= measure of EXTERIOR ANGLE

360°

24=15°

And the interior and exterior angles form a linear pair, so:


substracting measure of EXTERIOR ANGLE from both sides:

measure of INTERIOR ANGLE=180°-measure of EXTERIOR ANGLE

=180°-15°

So: measure of EXTERIOR ANGLE = 15°

measure of INTERIOR ANGLE=165° Standard 12



standard 12 basic polygon definitions ... - whs...

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